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Proc. R. Soc. Adoi:10.1098/rspa.2009.0348

Published online

On the thermodynamic entropyof fatigue fracture

BY M. NADERI, M. AMIRI AND M. M. KHONSARI*

Department of Mechanical Engineering, Louisiana State University,Baton Rouge, LA 70803, USA

Entropy production during the fatigue process can serve as a measure of degradation.We postulate that the thermodynamic entropy of metals undergoing repeated cyclic loadreaching the point of fracture is a constant, independent of geometry, load and frequency.That is, the necessary and sufficient condition for the final fracture of a metal undergoingfatigue load corresponds to a constant irreversible entropy gain. To examine validity,we present the results of an extensive set of both experimental tests and analyticalpredictions that involve bending, torsion and tension-compression of aluminium 6061-T6and stainless steel 304 specimens. The concept of tallying up the entropy generationhas application in determining the fatigue life of components undergoing cyclic bending,torsion and tension-compression.

Keywords: thermodynamic entropy; fatigue; cyclic loading

1. Introduction

All structures and machinery components undergoing fatigue loading are proneto crack formation (Bullen et al. 1953) and its subsequent growth that increaseswith time. When a crack is formed, the strength of the structure or the componentdecreases, and it can no longer function in the intended manner for which it wasdesigned. Moreover, the residual strength of the structure decreases progressivelywith increasing crack size. Eventually, after a certain time, the residual strengthbecomes so low that the structure fails (Broek 1982). It is, therefore, of paramountimportance to be able to predict the rate of decline in the component’s residualstrength and the remaining life of the system.

Many researchers have attempted to quantify fatigue in order to predict thenumber of cycles to failure. Among them, Miner (1945) pioneered the ideaof quantifying fatigue damage based on the hypothesis that, under variableamplitude loading, the life fractions of the individual amplitudes sum to unity.Later, Coffin (1971) and Manson (1964) independently proposed the well-knownempirical law �εp/2 = ε′

f(Nf)c that relates the number of cycles to failure, Nf ,

in the low-cycle fatigue regime to the amplitude of the applied cyclic plasticdeformation, �εp/2, for a material with given mechanical properties, ε′

f and c.

*Author for correspondence ([email protected]).

Received 4 July 2009Accepted 21 September 2009 This journal is © 2009 The Royal Society1

mailto:[email protected]

http://rspa.royalsocietypublishing.org/

2 M. Naderi et al.


The role of energy dissipation associated with plastic deformation during fatigueloading as a criterion for fatigue damage was also investigated by Halford (1966)and Morrow (1965).

The energy approach for estimating the fatigue life of materials under cyclicloading tests has gained considerable attention by researchers (Morrow 1965;Blotny & Kalcta 1986; Atkins et al. 1998; Fengchun et al. 1999; Park & Nelson2000; Gasiak & Pawliczek 2003; Jahed et al. 2007; Meneghetti 2007). Morrow’spaper (1965) is representative of a pioneering work that takes into account cyclicplastic energy dissipation and fatigue of metals that undergo cyclic loading.A descriptive theory of fatigue was presented that uses the cumulative plasticstrain energy as a criterion for fatigue damage and the elastic strain energy as acriterion for fracture. For fully reversed fatigue load, Morrow derived a relation forplastic strain energy per cycle Wp in terms of the cyclic stress–strain properties,applicable when plastic strain is predominant. Park & Nelson (2000) proposedan empirical correlation for the estimation of fatigue life, taking into account theelastic strain energy We as well as plastic strain energy Wp. In the high-cycleregime, plastic strains are usually quite small. Park & Nelson (2000) proposedthat the two energy terms, Wp and We, must be combined into the total strainenergy parameter Wt,

Wt = Wp + We = AN αf + BN β

f , (1.1)

where the constants A, α, B and β can be determined from a set of uniaxial fatiguetest data that cover a sufficiently large number of cycles. The energy dissipationowing to plastic deformation during fatigue is a fundamental irreversiblethermodynamic process that must be accompanied by irreversible entropy gain.

Permanent degradations are the manifestation of irreversible processes thatdisorder a system and generate entropy in accordance to the second law ofthermodynamics. Disorder in systems that undergo degradation continues toincrease until a critical stage when failure occurs. Simultaneously, with the risein disorder, entropy monotonically increases. Thus, entropy and thermodynamicenergies offer a natural measure of component degradation (Basaran & Yan 1998;Doelling et al. 2000; Bryant et al. 2008; Amiri et al. in press). Of interest in thispaper is to quantify the entropy rise in bending, torsion and tension-compressionfatigue of metallic components, and particularly the entropy at the instance whenfailure occurs. According to Whaley (1983), the entropy at the fracture pointcan be estimated by integrating the cyclic plastic energy per temperature ofmaterial. The hypothesis of this paper is that, at the instance of failure, thefracture fatigue entropy (FFE) is constant, independent of frequency, load andspecimen size.

2. Experimental procedure

A series of fatigue tests are performed to examine the validity of the proposedhypothesis. Three different stress states examined are completely reversedbending, completely reversed torsion and axial loads. Tests are conducted withaluminium (Al) 6061-T6 and stainless steel (SS) 304 specimens. The fatiguetesting apparatus used is a compact, bench-mounted unit with a variable-speed

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Thermodynamic entropy of fatigue 3


torsion fixture

specimen

IR camera

control box

Figure 1. Schematic of the experimental apparatus for torsion fatigue tests.

motor, variable throw crank connected to the reciprocating platen, with a failurecut-off circuit in a control box, and a cycle counter. The variable throw crankis infinitely adjustable from 0 to 50.8 mm to provide different levels of stressamplitude. The same fatigue apparatus is used for applying torsion, bending andaxial load using appropriate fixtures.

Figure 1 shows a schematic of the experimental setup used for torsion tests.The torsional fatigue tests are made using a round bar specimen clamped at bothends and rotationally oscillated at one of the ends via a crank with specifiedamplitude and frequency. Bending fatigue tests involve a plane specimen clampedat one end and oscillated at the other end, which is connected to the crank. Thetension-compression fatigue tests involve clamping a plate specimen at both endsin the grips and oscillating the lower grip at a specified amplitude and frequency.All tests are conducted by installing a fresh specimen in the apparatus, specifyingthe operating condition and running continuously until failure occurs. All testsare run until failure, when the specimen breaks into two pieces.

High-speed, high-resolution infrared (IR) thermography is used to record thetemperature evolution of the specimen during the entire experiment. Beforefatigue testing, the surface of the specimen is covered with a black paint toincrease the thermal emissivity of the specimen surface. Figure 2 shows thesurface-temperature evolution of a series of bending fatigue tests at the clampedend where the specimen fractures. These tests pertain to subjecting an Alspecimen to different stress amplitudes. It is to be noted that a persistent trendemerges from all the experiments. Initially, the surface temperature rises as theenergy density associated with the hysteresis effect gives rise to the generation ofheat greater than the heat loss from the specimen by convection and radiation.Thereafter, temperature tends to become relatively uniform for a period of timeuntil it suddenly begins to rise, shortly before failure occurs. Figure 2 also shows

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4 M. Naderi et al.


48

T (

°C)

40

a

b

cd

mean temperature

temperature fluctuationowing to thermoelastic effect

32

240 5000

number of cycles

10 000 15 000

Figure 2. Evolution of temperature in the bending fatigue of an Al specimen at 10 Hz at differentdisplacement amplitudes: (a) 49.53 mm, (b) 48.26 mm, (c) 38.1 mm and (d) 35.56 mm. Temperatureincreases initially, levels off for a period and suddenly rises just before fracture occurs.

how the temperature of the specimen varies around a mean value. The rise of themean temperature during fatigue tests is due to the plastic deformation of thematerial. The oscillation of the temperature around the mean value is caused bythe thermoleastic effect (Yang et al. 2001; Meneghetti 2007).

3. Theory and formulation

Description of the relevant irreversible processes requires formulating the firstand second laws of thermodynamic as applicable to a system whose propertiesare continuous functions of space and time. According to the first law ofthermodynamics, the total energy content E within an arbitrary control volumecan change only if energy flows into (or out of) the control volume throughits boundary

dE = dQ − dW , (3.1)

where Q and W are heat flow and work across the boundary of the control volume.In terms of the specific quantities, the law of conservation of energy for a controlvolume can be written as (de Groot & Mazur 1962)

ρdudt

= −divJ q + σ : D, (3.2)

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where ρ is the density, u is the specific internal energy, J q is the heat flux acrossthe boundary, σ is the symmetric stress tensor and D is the symmetric rate ofdeformation tensor.

The second law of thermodynamics (Clausius–Duhem inequality) postulatesthat the rate of entropy generation is always greater than or equal to the rate ofheating divided by the temperature T (Lemaitre & Chaboche 1990), i.e.

ρdsdt

≥ −div(J q

T

), (3.3)

where s represents the specific entropy. The right-hand side of equation (3.3) canbe written as

div(J q

T

)= divJ q

T− J q · grad T

T 2. (3.4)

Substituting equation (3.4) into equation (3.3) and replacing divJ q from equation(3.2) yields

ρdsdt

+ (σ : D − ρ du/dt − J q · grad T/T )

T≥ 0. (3.5)

Let Ψ represent the specific free energy defined as (Lemaitre & Chaboche 1990)

Ψ = u − Ts. (3.6)

Differentiating equation (3.6) with respect to time t, and dividing the resultby temperature T yields

−(dΨ/dt + s dT/dt)T

= dsdt

− du(T dt)

. (3.7)

Considering equation (3.7), the inequality (3.5) reads

(σ : D − ρ(dΨ/dt + s dT/dt) − J q · grad T/T )

T≥ 0. (3.8)

For small deformations, the deformation rate tensor D is replaced by ε, whichrepresents the total strain rate. The total strain is decomposed to plastic andelastic strain,

ε = εp + εe. (3.9)

The specification of the potential function (free specific energy Ψ ) must beconcave with respect to temperature T and convex with respect to other variables.Also, the potential function Ψ depends on the observable state variables andinternal variables (Lemaitre & Chaboche 1990),

Ψ = Ψ (ε, T , εp, εe, Vk), (3.10)

where Vk can be any internal variable.

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6 M. Naderi et al.


By referring to equation (3.8), strains are decomposed to ε − εp = εe, so we canrewrite equation (3.10) as

Ψ = Ψ (T , ε − εp, Vk) = Ψ (T , εe, Vk). (3.11)

Using the chain rule, the rate of specific free energy can be written as

∂Ψ

∂t=

(∂Ψ

∂εe

): εe +

(∂Ψ

∂T

)T +

(∂Ψ

∂Vk

)V k. (3.12)

After the substitution of equation (3.12) into equation (3.8), we obtain

((σ − ρ∂Ψ/∂εe) : εe + σ : εp − ρ(∂Ψ/∂T + s)T − ρ∂Ψ/∂VkV k−J q · grad T/T )

T≥ 0. (3.13)

For small strains, the following expressions define the thermoelastic laws(Lemaitre & Chaboche 1990):

σ = ρ∂Ψ

∂εe(3.14)

and

s = −∂Ψ

∂T. (3.15)

The constitutive laws of equations (3.14) and (3.15) arise from the fulfilment of thenon-negative inequality of equation (3.13). The thermodynamic forces associatedwith the internal variables (Lemaitre & Chaboche 1990) are defined as follows:

Ak = ρ∂Ψ

∂Vk. (3.16)

Hence, the Clausius–Duhem inequality is reduced to express the fact that thevolumetric entropy generation rate is positive,

γ = σ :εp

T− AkV k

T− J q · grad T

T 2≥ 0. (3.17)

Equation (3.17) is also interpreted as the product of generalized thermodynamicforces, X = {σ/T , Ak/T , grad T/T 2}, and generalized rates or flows, J ={εp, −V k, −J q}, (Prigogine 1967; Bejan 1988; Kondepudi & Prigogine 1998)

γ =∑

k

Xk · Jk. (3.18)

Irreversible thermodynamics consider forces X as the drivers of flows J . Each Jcan depend on all forces (de Groot & Mazur 1962) and intensive quantities (e.g.temperature T ) associated with the dissipative process.

Equation (3.17) describes the entropy generation process that consists ofthe mechanical dissipation owing to plastic deformation, non-recoverable energystored in the material and the thermal dissipation owing to heat conduction. Formetals, non-recoverable energy represents only 5–10% of the entropy generation

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owing to mechanical dissipation and is often negligible (Clarebrouhg et al. 1955,1957; Halford 1966),

AkV k

T≈ 0. (3.19)

Therefore, equation (3.17) reduces to

γ = σ :εp

T− J q · grad

TT 2

≥ 0. (3.20)

The coupling of thermodynamics and continuum mechanics requires the selectionof observable and internal variables (Basaran & Nie 2004). In the present study,two observable variables, temperature T and total strain ε, are chosen. Byreferring to equation (3.2) and replacing ρ du/dt by the expression derived fromu = Ψ + Ts,

ρ(Ψ + Ts + T s) = −divJ q + σ : D. (3.21)

Considering equations (3.12), (3.14) and (3.15) and small deformations,equation (3.21) yields

σ : εe + AkV k + ρTs = −div J q + σ : ε. (3.22)

By applying the chain rule to equation (3.15), we can express s by

s = − ∂2Ψ

∂εe∂T: εe − ∂2Ψ

∂T 2T − ∂2Ψ

∂Vk∂TV k. (3.23)

Substitution of equations (3.27), (3.15) and (3.16) into equation (3.23)results in

s = − ∂σ

ρ∂T: εe + ∂s

∂TT − ∂Ak

ρ∂TV k. (3.24)

By introducing the specific heat, C = T (∂s/∂T ), using equations (3.9), (3.19)and (3.24) and taking into account Fourier’s law (J q = −k grad T ), equation(3.22) leads to (Lemaitre & Chaboche 1990)

k∇2T = ρCT − σ : εp − T∂σ

∂T: εe, (3.25)

where k is the thermal conductivity.Equation (3.25) shows the energy balance between four terms: transfer of

heat by conduction (k∇2T ), retardation effect owing to thermal inertia (ρCT ),internal heat generation consisting of plastic deformation (Wp = σ : εp)—whichis responsible for mean temperature rise—and the thermoelastic coupling term,We = T ∂σ/∂T : εe, which takes into account the thermoelastic effect (figure 2).

The total energy generation in equation (3.21) is the combination of elasticand plastic energy, Wt = We + Wp for low- and high-cycle fatigue (Morrow 1965;Halford 1966; Park & Nelson 2000),

Wt = 2(1 + υ)σ ′2f N 2b

3E+ 4ε′

f(1−n′)(1+n′)σ

(1+n′)/na

σ′1/n′f

, (3.26)

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8 M. Naderi et al.


where n ′ is the cyclic strain hardening exponent, ε′f is the fatigue ductility

coefficient, σ ′f denotes the fatigue strength coefficient, σa represents the stress

amplitude and υ is Poisson’s ratio. The parameters b, E and N represent thefatigue strength coefficient, the modulus of elasticity and the number of cycles tofailure, respectively.

As the temperature fluctuation caused by thermoelastic effect is small incomparison with the mean temperature rise (figure 2), the elastic part in equation(3.25) can be neglected (Meneghetti 2007). Therefore, equations (3.20) and (3.25)can be simplified to

ρCT − k∇2T = Wp (3.27)

and

γ = Wp

T− J q · gradT

T 2≥ 0. (3.28)

The FFE can be obtained by the integration of equation (3.28) up to the timetf when failure occurs,

γf =∫ t f

0

(Wp

T− J q · gradT

T 2

)dt, (3.29)

where γf is the FFE. In low-cycle fatigue where the entropy generation owingto plastic deformation is dominant and the entropy generation owing to heatconduction is negligible, equation (3.25) reduces to

γf =∫ tf

0

(Wp

T

)dt. (3.30)

The experimental temperatures, such as those shown in figure 2, can be usedto calculate the FFE.

4. Numerical simulation

Simultaneous solution of equations (3.27) and (3.29) is necessary to determine theentropy generation. For this purpose, a commercial software package (FLEXPDE),which employs the finite-element method to solve partial differential equations,is used.

(a) Computational model

Three-dimensional models with 10-node quadratic tetrahedral elements andappropriate number of meshes for the specimens undergoing bending aredeveloped. The corresponding number of finite elements for bending is 2709.Figure 3 shows the geometry and finite-element meshes used for the specimenundergoing bending fatigue, and, because of the symmetric condition, only halfof the specimen is modelled.

A mesh dependency study was carried out to investigate the effect of thenumber of meshes on the calculated entropy generation from equation (3.29).The results of the effect of mesh refinement for the bending test of Al 6061 at10 Hz and 49.53 mm displacement amplitude are shown in table 1. It reveals thatthe calculated result for the FFE is independent of mesh refinement.

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YZ

X

Figure 3. Finite-element model and associated mesh in bending fatigue.

Table 1. Effect of mesh refinement on calculated FFE.

number of mesh FFE (MJm−3 K−1)

2709 3.9602897 3.9555604 3.956

10 771 3.95415 875 3.955

(b) Boundary conditions

Figure 4 shows a two-dimensional sketch of the computational model usedfor the bending load, with the notations indicating the boundary conditions.A summary of the boundary conditions is shown in table 2. Different tipdisplacement amplitudes (25–50 mm) at different frequencies (6–18 Hz) areconsidered as the applied loads in the model. Boundary W1 exchanges heat to thesurroundings by convection and radiation. Walls W2 are at room temperature, Ta.Convective heat transfer is assumed as the boundary condition on walls W3. Theconvective heat transfer coefficient h is estimated using an experimental procedurethat involves measuring the cooling rate of the specimen surface temperature aftera sudden interruption of the fatigue test (Amiri et al. in press). Surface emissivity,ε0, is calculated to be 0.93 and σ0 is the Stephan–Boltzmann constant that is equalto 5.67 × 10−8 Wm−2 K−4.

Walls W4 are associated with the glass-wool insulation used in the experiments,therefore, there is zero heat flux at this boundary. The boundary W5 is consideredas a symmetric boundary condition.

Thermal and mechanical properties of the materials are summarized in table 3(ASM 1990; Bejan 1993). Fatigue properties of the selected materials are basedon the experimental studies of Wong (1984) and Lin et al. (1992).

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10 M. Naderi et al.


ZW3

W3

W3

W3

W2

W2

W2

W2

W2

W2 W2

W2

W2W1

W1

axis of symmetry, W5

(a)

(b)

W1W4

W4

W4

Y

X

X

Figure 4. Schematic of the two-dimensional model with boundary condition notations: (a) top viewand (b) side view.

Table 2. Boundary conditions.

boundary type thermal condition description

W1 wall, convection and k∂T/∂n = h(T − Ta) + σ0 n is the normal to the wallradiation to air × ε0(T 4 − T 4

a )

W2 wall, constant T T = Ta —W3 wall, convection to air k∂T/∂n = h(T − Ta) n is the normal to the wallW4 wall, insulation ∂T/∂n = 0 n is the normal to the wallW5 wall, symmetric plane ∂T/∂n = 0 n is the normal to the wall

Table 3. Material properties.

material k(Wm−1 K−1) ρ(kg m−3) C (Jkg−1 K−1) σ ′f (Mpa) ε′

f n′

Al 6061 164 2659 871 535 1.34 0.095SS 304 16 7900 500 1000 0.25 0.171glass-wool 0.037 200 0.66 — — —

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0

number of cycles

0.5

1.5

volu

met

ric

entr

opy

gene

ratio

n (M

J m

–3 K

–1)

2.5

3.5

4.0

3.0

2.0

1.0

100 101 102 103 104

Figure 5. Volumetric entropy generation evolution versus the number of cycles for Al 6061-T6 underbending tests, frequency = 10 Hz, displacement amplitudes = 49.53 mm. Filled circle, experimental;solid line, simulation.

5. Results and discussion

The evolution of entropy generation is calculated for the entire fatigue lifeand then integrated over time to determine the entropy generated during thefatigue process (equation (3.29)). Figure 5 shows the comparison of numericaland experimental entropy generation based on equations (3.29) and (3.30) forthe bending fatigue of Al 6061-T6, where the frequency and displacementamplitude are 10 Hz and 49.53 mm, respectively. The small difference betweenthe experimental result and the numerical simulation is due to the fact thatheat conduction is neglected in equation (3.30). The final value of the entropygeneration (about 4 MJ m−3 K−1 for this test) is associated with the entropy atfracture when the specimen breaks into two pieces. An uncertainty analysis isperformed using the method of Kline & McClintock (1953). The maximum errorin calculating entropy based on uncertainty analysis is about ±1 per cent.

Figure 6 shows the results of experimental FFE for bending fatigue tests atdifferent frequencies. Results of different displacement amplitudes and differentthicknesses of specimen, i.e. 3, 4.82 and 6.35 mm, are shown in this figure. TheFFE is found to be about 4 MJ m−3 K−1, regardless of the load, frequency andthickness of the specimen. It is to be noted that the results of seven sets ofexperiments presented in figure 6 correspond to the different combinations ofspecimen thicknesses and operating frequencies. Also, the experimental data areassociated with the different displacement amplitudes ranging from 25 to 50 mm.The same concept for plotting the experimental data is followed in figures 7 and 8.

Figure 7 presents the results of experimental FFE plotted as a function of thefatigue life for bending and tension-compression tests for Al 6061-T6 specimensat 10 Hz. It is seen that the FFE is independent of the type of loading.

Figure 8 presents the results of entropy generation at failure for SS 304undergoing bending, and torsion fatigue tests. The results show that the entropygeneration at the fracture point for SS 304 is about 60 MJ m−3 K−1, independent

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12 M. Naderi et al.


number of cycles to failure103 104

frac

ture

fat

igue

ent

ropy

(MJ

m–3

K–1

)

0

1

2

3

4

5

5 × 102

Figure 6. Fracture fatigue entropy versus the number of cycles to failure for different bendingfatigue tests of Al 6061-T6 with different specimen thicknesses, frequencies and displacementamplitudes. Fracture fatigue entropy remains at roughly 4 MJ m−3 K−1, regardless of the thicknessload and frequency. Displacement amplitude is varied from 25 to 50 mm. Filled circle, thickness =6.35 mm, f = 10 Hz; filled diamond, thickness = 3.00 mm, f = 10 Hz; filled star, thickness = 4.82 mm,f = 10 Hz; unfilled circle, thickness = 6.35 mm, f = 6.5 Hz; unfilled triangle, thickness = 4.82 mm,f = 12.5 Hz; unfilled star, thickness = 6.35 mm, f = 6.5 Hz; unfilled diamond, thickness =6.35 mm, f = 12.5 Hz.

number of cycles to failure

104

frac

ture

fat

igue

ent

ropy

(MJ

m–3

K–1

)

0

1

2

3

4

5

2 × 103 3 × 104

Figure 7. Experimental fracture fatigue entropy versus the number of cycles to failure for tension-compression, bending and torsional fatigue tests of Al 6061-T6 at frequency 10 Hz. Fracturefatigue entropy remains at about 4 MJ m−3 K−1 for both tension-compression and bending fatigue.Displacement amplitude is varied from 25 to 50 mm. Filled square, tension-compression; filled circle,bending; filled star, torsion.

of frequency and geometry. It is to be noted that the fatigue life of a specimenundergoing a cyclic load is only weakly dependent on the test frequencies (Morrow1965; Liaw et al. 2002) up to 200 Hz.

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number of cycles to failure

104

frac

ture

fat

igue

ent

ropy

(MJ

m–3

K–1

)

0

10

30

20

40

60

50

70

7 × 103 2.5 × 104

Figure 8. Experimental fracture fatigue entropy versus the number of cycles to failure for bendingand torsional fatigue tests of SS 304 for different loads (25–50 mm displacement amplitudes) andfrequencies. Fracture fatigue entropy remains at about 60 MJ m−3 K−1 for tension-compression andbending and torsion fatigue. Filled circle, bending, f = 10 Hz; filled triangle, bending, f = 18 Hz;filled diamond, bending, f = 6 Hz; filled star, torsion, f = 10 Hz.

The results presented in figures 6–8 demonstrate the validity of the constantentropy gain at failure for Al and SS specimens. The results reveal that thenecessary and sufficient condition for final fracture of Al 6061-T6 corresponds tothe entropy gain of 4 MJ m−3 K−1, regardless of the test frequency, thickness of thespecimen and the stress state. For SS 304 specimens, this condition correspondsto an entropy gain of about 60 MJ m−3 K−1.

A possible application of the proposed hypothesis of the constant entropygain at failure is in the development of a methodology for the prevention of thecatastrophic failure of metals undergoing fatigue load. As demonstrated in thiswork (figure 5), the entropy generation increases during the fatigue life towards afinal value of γf . Thus, the FFE can be used as an index of failure. As the entropygeneration accumulates towards the FFE, it provides the capability of shuttingdown of the machinery before a catastrophic breakdown occurs.

The concept of constant entropy gain at the fracture point, γf , assumes thatthe thermodynamic condition associated with the entropy generation is identicalduring the fatigue process and varies only in the duration of the process, i.e.failure occurs when

N = Nf and γ = γf . (5.1)

Within the range of the experimental tests presented, γf is only dependent uponthe material and is independent of load, frequency and thickness. Therefore, theduration of the fatigue process varies depending on the operating conditions inorder to satisfy the condition of equation (5.1).

Based on this concept, one can conduct an accelerated failure testing schemeby increasing process rates J while maintaining equivalent thermodynamic forcesX to obtain the same sequence of physical processes, in identical proportions,but at a higher rate. For example, by increasing frequency, the rate of plastic

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14 M. Naderi et al.


1.0

0.8

0.6

0.4

0.2

0 0.2 0.4 0.6 0.8 1.0

normalized number of cycles, N / Nf

norm

aliz

ed e

ntro

py g

ener

atio

n, γ

/ γ f

Figure 9. Normalized entropy generation versus normalized number of cycles for bending fatigueof SS 304 and Al 6061-T6 for different thicknesses of specimen, displacement amplitudesand frequencies. Filled circle, Al, thickness = 4.82 mm, f = 10 Hz, displacement amplitude =38.1 mm; filled square, Al, thickness = 6.35 mm, f = 10 Hz, displacement amplitude = 30.5 mm; filledtrinagle, SS, thickness = 3 mm, f = 10 Hz, displacement amplitude = 48.2 mm; filled diamond, SS,thickness = 3 mm, f = 16 Hz, displacement amplitude = 45.7 mm.

deformation εp increases, and subsequently the rate of degradation increases whilethe duration of the test is shortened in order to satisfy equation (5.1). This isin accordance with the accelerated testing procedure recently put forward byBryant et al. (2008) based on the thermodynamics of degradation.

Figure 9 shows the normalized entropy generation during the bending fatigueof SS 304 and Al 6061-T6 for different thicknesses, displacement amplitudesand frequencies. The abscissa of figure 9 shows the entropy generation usingequation (3.29) and normalized by dividing the entropy gain at the final fracture,γf . The ordinate shows the number of cycles normalized by dividing the finalnumber of cycles when failure occurs. It can be seen that the normalized entropygeneration monotonically increases until it reaches the entropy at the failurepoint. Interestingly, a similar trend between normalized wear plotted againstthe normalized entropy was reported by Doelling et al. (2000). Their workresulted in the prediction of flow of Archard’s wear coefficient (Archard 1953)with remarkable accuracy.

The relation between the normalized cycles to failure and the normalizedentropy generation is approximately linear and can be described as

γ

γf

∼= NNf

, (5.2)

where γf is a property of the material. Using equation (5.2), the number of cyclesto failure can be expressed as

Nf∼=

(Nγ

)γf . (5.3)

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Equation (5.3) offers a methodology for prediction of the fatigue failure of a givenmaterial based on the measurement of the thermodynamic entropy generation.By having FFE (or γf ) and calculating entropy generation γ at a selected numberof cycles N , the fatigue life Nf of the specimen can be predicted. Furthermore, anaccelerated testing method can be developed whereby one calculates the entropygeneration γ over the first few cycles and determines Nf .

6. Conclusions

A thermodynamic approach for the characterization of material degradation isproposed, which uses the entropy generated during the entire life of the specimensundergoing fatigue tests. Results show that the cumulative entropy generationis constant at the time of failure and is independent of geometry, load andfrequency. Moreover, it is shown that the FFE is directly related to the typeof material. That is, materials with different properties, such as SS and Al have adifferent cumulative entropy generation at the fracture point. Within the range ofconditions tested, the results show that the entropy generation is approximately4 MJ m−3 K−1 for Al 6061-T6 and 60 MJ m−3 K−1 for SS 304. The implication ofthis finding is that, by capturing the temperature variation of a system undergoingfatigue process, the evolution of entropy generation can be calculated during thefatigue life and then compared to the appropriate FFE for the material to assessthe severity of the degradation of the specimen. Also, a methodology is offered forthe prediction of the fatigue failure of a given material based on the measurementof the thermodynamic entropy generation.

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on the thermodynamic entropy of fatigue...

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